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Simplifying 12k2 + -13k + -3 = 0 Reorder the terms: -3 + -13k + 12k2 = 0 Solving -3 + -13k + 12k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.25 + -1.083333333k + k2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.083333333k + 0.25 + k2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.083333333k + k2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.083333333k + k2 = 0 + 0.25 -1.083333333k + k2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.083333333k + k2 = 0.25 The k term is -1.083333333k. Take half its coefficient (-0.5416666665). Square it (0.2934027776) and add it to both sides. Add '0.2934027776' to each side of the equation. -1.083333333k + 0.2934027776 + k2 = 0.25 + 0.2934027776 Reorder the terms: 0.2934027776 + -1.083333333k + k2 = 0.25 + 0.2934027776 Combine like terms: 0.25 + 0.2934027776 = 0.5434027776 0.2934027776 + -1.083333333k + k2 = 0.5434027776 Factor a perfect square on the left side: (k + -0.5416666665)(k + -0.5416666665) = 0.5434027776 Calculate the square root of the right side: 0.737158584 Break this problem into two subproblems by setting (k + -0.5416666665) equal to 0.737158584 and -0.737158584.Subproblem 1
k + -0.5416666665 = 0.737158584 Simplifying k + -0.5416666665 = 0.737158584 Reorder the terms: -0.5416666665 + k = 0.737158584 Solving -0.5416666665 + k = 0.737158584 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + k = 0.737158584 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + k = 0.737158584 + 0.5416666665 k = 0.737158584 + 0.5416666665 Combine like terms: 0.737158584 + 0.5416666665 = 1.2788252505 k = 1.2788252505 Simplifying k = 1.2788252505Subproblem 2
k + -0.5416666665 = -0.737158584 Simplifying k + -0.5416666665 = -0.737158584 Reorder the terms: -0.5416666665 + k = -0.737158584 Solving -0.5416666665 + k = -0.737158584 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + k = -0.737158584 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + k = -0.737158584 + 0.5416666665 k = -0.737158584 + 0.5416666665 Combine like terms: -0.737158584 + 0.5416666665 = -0.1954919175 k = -0.1954919175 Simplifying k = -0.1954919175Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.2788252505, -0.1954919175}
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